Tracking the Partially Illuminated Earth or Moon*

TRACKING TIlE PARTIALLY ILLUMINATED EARTII OR MOONk by Dr. Roger S. Estey Chief, Radiometry Group Northrop Space Laboratories Hawthorne, California ABSTRACT

The planet tracking instrument comprises a variable focus telescope, scanning mechanism to sense the planetary disk correctly, gimbals which are articulated about three axes, and a computer for reducing the data and for generating commands to the tracker servo.

The observation of the sightline to the center of the partially illuminated earth or moon is accomplished by a series of measurements by which the circular limb of the body is centered concentrically with the circular field stop in the instrument. Since the limb must be distinguished from the terminator independent knowledge of the sun's direction is required to align the tracker mechanism appropriately. The measurements and calculations are digital and provide servo commands to align the gimballed tracker precisely on the geometrical center of the celestial body.

CHARACTERISTICS OF TIlE PLANETARY TARGETS

INTRODUCTION The navigation of spacecraft within the solar system requires the observation of the line of sight to distant celestial bodies (stars) or to neighboring bodies (the sun, moon or planets). With the exception of the sun these near bodies exhibit phase and consequently an observed line of sight may be ambiguous depending on whether the line of sight terminates at the centroid of luminosity or the geometrical center of the planetary disk. The present paper discusses automatic instrumentation for aligning the optical axis of an instrument in a spacecraft with the geometrical center of a nearby partially illuminated celestial body. Although the concepts and instrumentation to be described are applicable to any planet or satellite, the discussion will be directed to observations of the earth or moon. At great ranges the angular subtense of the planetary disk is so small that the angular difference in pointing direction between the luminous centroid and the center of the circular disk may be less than the angle error due to other causes and hence negligible. At short range the angular subtense of the body is so large that special optical or mechanical scanning techniques are required. The discussion which follows will emphasize midcourse ranges in which the earth or moon subtends about twenty degrees or less. At the earth-moon distance the moon subtends half a degree and its cent er can be observed to about 12 seconds by the techniques described here. This accuracy is far greater than the disparity between lines of sight to the luminous centroid or to the geometrical center. Thus in cis-lunar space the partial illumination problem is serious and a solution must be found if intolerable angle errors are to be avoided.

*

The trajectory of a spacecraft traveling between the earth and the moon will be used as an example to indicate the nature of the sighting and tracking problem which the instrument must solve. The various considerations which influence sightline observations in cis-lunar space will be discussed under two major headings: the sightline geometry and the properties of the celestial bodies concerned. Sight line Geometry The lunar orbit is inclined approximately 5 degrees to the plane of the ecliptic and the intersection of the ecliptic and orbital planes is located at the earth; consequently a sightline to the sun (at 92,000,000 miles) has a maximum deviation from the ecliptic plane of only 0.01 degree, which is negligible for most purposes. The most useful set of coordinates for lunar navigation is a right-handed orthogonal set built up on the first point of Aries on the Y-axis and the north ecliptic pole on the Z-axis. In this set a position point has the coordinates A, celestial longitude, and ~, celestial latitude. The Nautical Almanac, planned for the convenience of terrestrial navigation and earth-based astronomical studies, presents the lunar ephemerides in terms of the north terrestrial pole and equator, and the meridian circle through Greenwich. Conversions between the various coordinate systems are not difficult but may be laborious; consequently, digital computer routines are frequently employed for this purpose. The angles subtended at the spacecraft by the earth or moon have been computed from the diameters of the bodies considered as spheres and using the values 7926 and 2157 miles respectively. Except for regions in close proximity to either body the subtended angles versus range are shown in Fig. 1 and in Table 1. Lunar trajectory studies have emphasized the need for midcourse corrections at about 90,000 miles and 20,000 miles from the moon. The instrument and technique reported here are adapted to this sort of midcourse navigation.

Presented at the IFAC Symposium on "Automatic Control in the Peaceful Uses of Space", Stavanger. Norway. June 21- 24. 1965. 355

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Fig. 1. Angular subtense or earth and moon.

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Fig. 2. Geometry of planetary phases.

Fig. 3. Spiral and radial slit reticles.

356

During an earth-moon voyage the lunar tracking problem must be considered to be dynamic in character. The significant angles, angular rates and ranges are time varying over the period of a lunar voyage and typically the rates increase rapidly as the lunar surface is approached. Although an exhaustive study of these variables is not feasible or necessary at this time, limited data are desirable background information for designing and appraising planetary tracking instruments and techniques.

body is a function of the sun-body-tracker geometry. The discussion which follows relates to the special case in which the moon is observed from a tracking station on earth. The more general cases involving tracking of earth, moon or other planet from a spacecraft are readily inferred. The observation of phase changes from earth is described in the literature. Lunar coordinate values of the position angle of the axis and the position angle of the bright limb are tabulated daily and reported in the American Ephemeris and Nautical Almanac, the section entitled "Ephemerides for Physical Observations". With the exception of narrow intervals at new or full moon where observations are difficult because of reduction in brightness and for other reasons the values are within a few degrees of 90 degrees or 270 degrees. This fortunate circumstance facilitates the alignment of a planet tracker which employs the techniques reported here.

A 42-hour one-way trajectory was available and forms the basis for the following discussion. In the intermediate range slower trajectories or circumlunar trajectories would yield closely similar conclusions. Rates derived from this trajectory are shown in Table 1. The phase of the earth or moon is identified by the angular relationship between sun, earth or moon, and the observer. The geometry is shown in Fig. 2. Sunlight incident in the direction SM illuminates the left hemisphere of the body as shown. By projection it can be shown that an observer on the line ME will see the partially illuminated body as a crescent disk in which L is the midpoint of the circular limb and T is the midpoint of the elliptical terminator. The phenomenon of phase is described by two terms, "phase" or "phase angle". The former relates to the relative luminous area, sometimes expressed as the relative length of the segment LT. Phase angle is the angle SME denoted bya. The relationship between these concepts is expressed by the equation

The Moon as a Target The radiometric characteristics of the earth and moon will be considered separately. Optical radiation from the moon is a blend of thermal radiation emitted by the body as a consequence of its temperature and of solar radiation reflected from the illuminated hemispherical portion of the lunar surface. The infrared signal derives from the temperature of the lunar surface which represents the balance between the incident radiation from the sun and the outward flow of radiation to space, modified by the effect of the heat capacity, conductivity, and the temperatures of material layers adjacent to the lunar surface. The range of temperature between the center of the fully illuminated disk and the dark limb has been reported by Pettit and Nicholson l as 342°K and l20 o K, respectively. Studies of the moon discussed by Ehricke 2 and Fielder 3 , indicate that the lunar surface is porous, of low heat conductivity, and that the surface temperature responds rapidly to changes in solar irradiation.

Phase = LT/LE = (1 + cos a)/2. As the illumination changes from new moon, for example, through first quarter to full, the phase changes from zero, through one-half, tu unity. The phase angle changes from 180 degrees through 90 degrees to 0 degrees in the same sequence. The tracking technique reported here requires measurements which are rather precisely related to the midpoint of the bright limb of the body. In order to align the tracking instrument the bright limb position angle must be known as an operating parameter. The orientation of the figure of the partially illuminated

The thermal radiation from the warmer portion of the lunar surface is considerable and is peaked at about ten microns. The radiance of the cold surface (the dark limb) is very small and is peaked near twenty-three microns. As the phase

TABLE 1 EARTH-MOON TRAJECTORY PARAMETERS Earth

Moon Elapsed Time (hr) 0 6 12 18 24 30 36 42

Range to Moon (m!) 5 2.38 x 10 1.83 1.48 1.15 0.85 0.55 0.30 0.0

Range Rate (mph) 9.5 x 10 5.5 5.3 5.0 4.6 3.7

3

Subtended Angle (deg) 0.32 0.68 0.84 1.07 1.46 2.25 4.13

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Angle Rate (deg/hr) 0.023 0.023 0.030 0.048 0.080 0.20 0.43

Subtended Angle (deg)

Angle Rate (deg/hr)

8.24 5.04 3.69 2.96 2.48 2.18 1.91

0.84 0.65 0.31 0.20 0.12 0.10 0.09

progresses from full moon through gibbous to crescent, the terminator moves back from the dark circular limb. The thermal gradient is steep right at the dark side of the terminator, but over the rest of the dark area the lower surface temperature is well established. In the shorter wavelength region near seven microns which is readily accessible by photoconductive detectors and where the signal from the warmer surface can be measured effectively, the signal from the colder area (dark limb) is about 3000 times less. In the longer wavelength region accessible only to relatively insensitive bolometers, the signals are less intense but the contrast between bright and dark limbs is reduced to values ranging from about eighty to one down to about twelve to one depending on the experimental conditions. Infrared is not a promising technique for the detection of the dark limb of a planetary body without atmosphere. Short wave radiation received from the moon is composed almost entirely of reflected sunlight. The contribution of starlight is negligible and the contribution of sunlight reflected from the earth is too small a signal to be useful. Re-' flected solar radiation has a distribution similar to that of solar energy but is modified by the reflectance of the lunar surface. The useful optical pass band is limited to the interval from about 0.4 to 0.65 microns by the properties of the most useful optical materials and the most sensitive photodetectors available. Within this pass band spectral variations in the reflectance of the terrain are not significant. For the moon (in terms of visible light) the albedo is 0.070 and the average reflectance is 0.106 2 ,3. From the position of a spacecraft between the earth and moon, when one body appears full, the other appears new, and vice versa. Using the terminology familiar to astronomers we will discuss the apparent visual magnitude as the observed illumination at the instrument caused by luminous flux from the moon. These data assume no intervening atmosphere. This illumination, E, expressed for example in lumens/m 2 , is related to ~ by the equation -0.4my

=

log E - log Eo

where ~ and E are the magnitude of and the illuminance produced by the body. Eo has the value 2.43 x 10- 6 lumens m- 2 and is the illuminance produced just outside the earth's atmosphere by a zero-magnitude star. 4 The luminance, B, is expressed in candles/m2 and depends on the phase and the surface area concerned. The quantity B is independent of the range to the observer, and to the degree that the lunar surface obeys Labert's cosine law, this luminance is independent of the angle of observation. Photometric information presently available concerning the moon comprises magnitude and phase law data from which either the total flux or the average brightness can be expressed as a function of phase angle. The significant numerical data are as follows: 5

Star outside earth's atmosphere, mv = 0, illuminance = 2.43 x 10- 6 Lux. Moon at mean opposition, a = 0, My = -12.70, illuminance outside earth's atmosphere, E = 0.292 lumens/m2 (Lux). Luminous intensity of full moon, I = 4.31 x 10 16 candles. The luminance of the moon does not vary with range but does vary with phase in a manner which depends on the incident solar radiation and on the luminous projected area. Casual inspection of th full moon indicates, and careful measurements J confirm, that the full moon is almost uniformly bright with a narrow zone of greater luminance at the limb. The average luminance of the partial moon falls off with phase as described above, and the cusps decrease in luminance even more rapidly than the central portion. Experiments performed at the Northrop observatory with the tracking equipment described in this paper have disclosed that for the crescent moon, the luminance at points on the limb 60 degrees from the midpoint is only half that at the midpoint. Furthermore, the luminance decreases from limb to terminator in a manner which is more pronounced towards the cusps as compared with measurements made near the center of the bright limb. The discussion thus far has tacitly assumed the projection of the moon to be a truly circular disk. The actual departures from true circularity are half a per cent of the radius or less. This quantity is trivial unless extreme pointing accuracy is required on a disk subtending a large angle. 6 7

The Earth as a Target '

As in the case with the moon, terrestrial radiation within the limits of the optical spectrum is a blend of emitted thermal radiation and reflected solar radiation. Infrared is emitted from the terrain and from the atmosphere to a degree depending on local temperature. The temperature range from equator to poles which exceeds to a considerable degree the temperature range between day and night controls the signal dynamic range to which an infrared limb sensing instrument must be designed. This temperature range is about -55°F to +llO°F. The corresponding range of blackbody radiation intensity, peaking near 9 to 12 microns, is about four to one. However, although the infrared region is effective in sensing the terrestrial limb, either light or dark, it is much less effective in observing bodies without a favorable atmosphere and consequently the instrument which operates in the visual region to sense the bright limb only is considered a preferable instrument because of its universal applicability. Short wave radiation received from the earth is composed entirely of reflected sunlight. The effect of moon and star light, aurorae, and air glow, is completely undetectable. Reflected

358

minimum ranges at which the errors will have specified values or less are tabulated below in Table 3.

solar radiation has a spectral distribution similar to that of the incident solar energy modified by the reflectance of the earth (land or sea) and the cloud layer. Within the useful optical pass band, 0.4 to 0.65 microns, the reflectance varies from 0.03 (ocean) to 0.86 (ice fields). There are moderate variations with wavelength as the unsaturated colors range from blue, through green and yellow to reddish brown. The visual albedo of the earth is 0.36~

TABLE 3 Sighting Errors Due to Non-Sphericity of Earth Minimum Range, mi. 3 2.1 x 10 4 2.1 x 10 5 1.3 x 10 6 1.3 x 10

Like the case of the moon, the brightness and the total flux from the earth vary with phase angle. The phase laws for the two bodies are substantially complementary, that is the earth is full when the moon is new and vice versa. For the full earth, the visual magnitude as seen from the sun is -3.80 and is as seen from the

min min sec sec

Analytically speaking, all methods for identifying and tracking the partially illuminated earth or moon fall into one of two classes. The first, or area-sensing technique bounds the planetary image with a closely matched circular field stop and compares the unvignetted area of the luminous disk with the reference area in the instrument; the lunar flux passing unobstructedly through the instrument is the basic element to be measured. If the circular portion of the planetary image is tangent to a circular aperture in the instrument, then any transverse movement of the planetary image with respect to the aperture will produce a reduction in the common area. Obviously, the sensitive portion of the total area is an annular region bounded by the instrument aperture; the area variations close to the null condition occur in this region. Consequently the balance of the total aperture area can be blanked out since it makes no contribution to the information.

Illuminance at moon due to full earth 12.31 lu/m2 Luminous intensity of full earth, I = 182 x 10 16 candles. Undoubtedly the luminance of the earth varies with phase in a manner similar to the brightness variation of the moon. In addition the brightness will vary along the terrestrial limb due to the varying reflectance of cloud cover and of terrain. Since the available signal flux is very large, compared to that from the moon for example, the variation in flux due to clouds, variable reflectance of terrain, or phase is rather small and is relatively unimportant. The discussion thus far has treated the projection of the earth as a truly circular disk, even though partially illuminated. In practice the terrestrial bright limb departs from true circularity due to the typical elevation of the cloud cover and of the terrain and the oblateness of the earth. In trajectories between celestial bodies within our solar system the spacecraft will travel in or close to the plane of the ecliptic. Consequently the effect of oblateness is maximized and the effect of mountainous terrain may be considerable. Typical elevations of these features are listed below in Table 2. TABLE 2

This technique is unsatisfactory because of large differences in angle sensitivity in various azimuths measured with respect to the line of cusps. Furthermore the considerable variation in luminance of different regions of the bright limb militates against this or other photometric technique for establishing the geometrical center of the celestial body. The second, or radial measurement technique, distinguishes the bright limb from other portions of the lunar contour, and by linear measurements of position locates the lunar image precisely with respect to reference coordinates in the instrument. Since the bright limb is a circular arc, but the terminator is an elliptical arc of variable eccentricity, linear measurements of the limb relative to the instrument axis appear to be an attractive means for determining the circle's center.

Earth Features Referred to Mean Radius Sphere Mean radius = 6,371 km. Feature

10 1 10 1

SURVEY OF TRACKING SCHEMES

moon.

Polar, sea level Polar, mountainous Mid-latitude, mountainous Mid-latitude, clouds Equatorial, sea level

Maximum Sight line Error

Altitude, km -14 -11 +10 +10 + 7

The offsets between the limb and the instrument axis can be measured, the center found from the equation of the circle and the tracking servos driven from the sightline errors computed along the gimbal axes. In a second approach emphasized in this report, the data receive minimum processing and are nul led in the servos without solving the circle equation explicitly.

From the data in the above table it is evident that limb measurements may lead to errors in the estimation of the disk center of about 10 km or 1:1200 of the disk diameter. At shorter ranges the sighting errors due to the nonsphericity of the earth are considerable. The 359

Boresighted Telescope

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Elevation Servo

Azimuth Servo

Azimuth Servo Amplifier Sensor Signal

Azimuth Readout

Data Processing Elevation Servo Amplifier

Elevation Readout

Fig. 4. Moon tracker block diagram .

...-

Timing Pulse Pick-Off

. - - - - - - Permanent Hagnet

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y-~'---'\!» ' Fig. 5. Opto-mechanical scanner system diagram.

360

•

Spiral Reticle Technique Even a cursory examination of the radial measurements approach suggests the simplification in which radial segments between the lunar image and the circular field diaphragm of the instrument are examined. If these segments are identical in length, the lunar image is concentric with the field diaphragm and the instrument axis coincides with the sightline to the geometrical center of the moon. There is a single restriction: namely, the observations must be limited to the circular limb of the moon.

Northrop equipment for sensing and tracking the geometrical center of the partially illuminated moon consists of a gimbaled pointing instrument and a separate electronics and computer package. The system, which is shown as a block diagram in Fig. 4, comprises a) an optical telescope, b) an automatic measuring device to observe the centration of the lunar image in the field of the instrument, c) a computer to calculate gimbal angle errors and generate servo commands, d) gimbal axes and servos, e) a finder telescope for reference, and f) angle read out devices on the gimbals. The optical pointing equipment generates the lunar limb measurements. Programming means combine with a computer to calculate the gimbal errors from equations of the form

A many-look approach for processing redundant data was investigated. However the three-look approach described here is more readily reduced to practice. In this technique the radial segments between the rim of the instrument and the moon's limb are determined at three places 60 degrees apart and approximately centered between the cusps. This technique was selected for breadboarding because of its adequacy and simplicity.

- (RC + R )/2 A

E

0.577 (RC

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The moon-sensing part of the breadboard system is an optomechanical instrument mounted in gimbals. The main housing contains the optics, the sensor, the preamplifier, and the motor-driven scanning subsystem. These details are shown in the exploded view, Fig. 5. The telescope objective is a custom designed and built telephoto lens having a 30 in. eqUivalent focal length, F/20 relative aperture and 2.3 degree angular field. There is a beam splitter in the optical path to permit the use of a separate reticle and eyepiece to form a boresighted telescope which monitors the tracking accuracy when tested against a simulated laboratory target or tested in the observatory against the natural moon. The optical layout includes a simple field lens assembly to image the entrance pupil of the objective into the aperture of the photosensor. The scan mechanism consists fundamentally of the fixed and radial slits and a rotating mask. The radial slits are 0.34 minutes wide and 60 degrees apart. The spiral slit is 0.17 minutes wide and covers its operating range in 270 degrees at 30 rps. The linear velocity of the aperture formed by the intersection of the radial and spiral slits is approximately 2500 arc-minutes per second of time in the radial direction. Each radial slit is sensed in 25 milliseconds. A shutter rotating at 4.29 rps permits the sensing of only one slit at a time. The rotating parts are driven at the appropriate speeds by a synchronous motor and gear train. When the instrument is aligned with respect to the moon, the three radial slits a, band c are approximately centered on the bright limb. Since the cusps of the waning and waxing moon face in opposite directions, a second set of slits on the other side of the reticle is provided (not shown in the figures). A supplementary shutter, hand-operated, permits either slit group to be exposed, depending on the moon's orientation at the time.

As the rotation of the spiral proceeds, the scanning aperture moves inward and in due course intersects the bright limb, producing a second abrupt voltage signal. Since the radial scan has constant linear velocity, the time between signals is an exact measure of the distance between the rim of the instrument (and hence the optical axis) and the bright limb of the body. The radial segment distances associated with slits A, Band C can be used to determine the sightline errors EX and E y by the equation B

2R

Design of Tracker Instrument

Consider an opaque reticle with three transparent radial slots disposed 60 degrees apart and a second opaque reticle with a transparent involute spiral slit as shown in Fig. 3. These reticles are almost in contact in the focal plane containing the planetary image. As the spiral rotates counter-clockwise, the intersection with a fixed slit, B, for example, will travel inward at constant linear velocity along the radius B. Since portions of the spiral intersect slits A and C, the operation with respect to B must be isolated by an appropriate aperture in a separate slowly rotating mask. When the radial slit on the spiral coincides with a radial slit in the fixed reticle, planetary energy anywhere in that part of the field will pass through the two coincident slits and generate a large voltage signal in the photocell which coincides exactly with the start of the linear radial scan.

2R

(X

= EX

0.577(R - RA) = E y c where R , R , and Rc are the radial intervals B associated with the slits A, B, and C, respectively. The constant 0.577 is an empirical scaling coefficient which balances the servo. DESCRIPTION OF A THREE-SPOKE BREADBOARD A breadboard model emphasizing the spiral scan and the three-look approach to data processing was designed, built and tested to demonstrate the effectiveness of the techniques just discussed by means of lunar observations from earth.

At one point in the scan system rotation, the radial aperture of the spiral reticle coincides with a fixed slit thus passing an intense pulse of moonlight. As the aperture moves along the radial

361

Fig. 6. Tracker unit showing partial disassembly.

Fig. 7. Electronics unit.

362

slit, the image of the lunar limb is reached and a second voltage pulse is produced. The spacing between these pulses is the radial distance between instrument rim and lunar limb at one of the slits a, b, or c. The radiation is sensed by a small 10-stage photomultiplier of commercial type. The signal voltage data from the photomultiplier are processed and error signals for the servos are computed. The gimbals have a range of ~ 10 degrees from their mean position. Each gimbal shaft, mounted on preloaded ball bearings, carries a worm sector of adequate range, which, in cooperation with a single-lead worm, gives a reduction of 432:1. These worm and sector assemblies, developed by Northrop in 1947, are of super-precision quality and have a proven accuracy of 5 arc seconds across the gear mesh. The servo motor and associated train are geared to the worm shaft. The same worm shaft gear drives a ten-turn precision potentiometer through additional gearing such that the 20-degree rotation of the gimbal shaft (limited by stops) corresponds to about 9-1/2 turns of the potentiometer. A potentiometer with a resolution of 0.03%, which is commercially available, will read 20 arc-seconds anywhere in the 20-degree range. The scanning process was critically examined with respect to various sources of error, including errors in determination of moon radii, Schmitt trigger drift, reticle eccentricity and misalignment, and the error over the tracking range. Since these errors are considered random, they have been combined, and the RMS error due to all of these causes is only 6.8 arc-seconds. Design of Computer and Data Processor The various electronic circuits associated with the lunar tracker are, except for the preamplifier, contained in a single electronics cabinet. The functions are discussed in terms of signal shaping and data processing. The earlier discussion has indicated how two successive pulses of radiant flux from the lunar image identify each of the radial increments from the bright limb to the rim of the instrument's field of view. Each of these pulses is a positive-going step up from the noise. At a selected reference level a Schmitt trigger fires an auxiliary gating circuit in a manner such that a measured train of clock pulses is gated into the computer. The number of clock pulses in the train is a measure of the radial increment R , R , or R as the case may be. Auxiliary circuit~ wigh sUit~ able time constants are used to deenergize the trigger circuit except within the interval expected to contain the measuring signal, thus minimizing the exposure to false signals due to no~se spikes which extend above the normal level of noise into the region monitored by the Schmitt trigger. The computer and associated circuitry are responsible for a) extracting the radial distances R , R , R from the signal output of the photo b m~ltiplie~, and b) computing error signals from

363

the equations

EX

2R b - (Ra + Rc)

- Ra) c The tracker computer is a transistorized digital system containing diode logic which performs addition and subtraction and provides storage in a digital-to-analog converter unit. The input signal is a digital series of clock pulses; output is analog voltage proportional to the solutions of the two equations. Fabrication of Experimental Model Tracker Ey

0.577(R

When fabricated, the experimental tracker was required to demonstrate the effectiveness of the tracking principle by means of a simple, reliable, inexpensive breadboard instrument. As built and delivered the breadboard comprises two units. The tracker unit including finder telescope weighs 82 pounds, and is designed for mounting on a pair of external equatorial axes. The electronics unit comprises signal processors, computer, servo amplifiers, and all necessary power supplies rackmounted in a cabinet 17 x 22 x 31-1/2 inches. The approximate weight is 150 pounds. Pictures of the tracking and electronics packages are shown in Figs. 6 and 7. In the fabrication of the opto-mechanical tracker unit techniques appropriate to the economical production of a single precision instrument were employed. Many subassemblies were provided with polished pads to facilitate alignment with autocollimators. Fits were achieved by closetolerance machining followed by lapping. These procedures avoided costly fixtures and assured accurate and stable alignment. The alignment of the reticle system was critical to provide a quiet gear train and to minimize the air gap between the two reticle elements. The gear train spacing was adjusted to an optimum value and the backlash was removed by compensation with a suitable spring. The alignment of the two conjugate reticle systems, the scanning system and the eyepiece system was within a few seconds tolerance in order not to deteriorate the boresight accuracy. With the alignment just described, if the real or laboratorysimulated lunar image is concentric with the boresight reticle pattern and the tracking error signal outputs indicate zero (i.e., no residual difference exists between the three radial scan line segments), then indeed the tracking system is pointed to the geometrical center of the partially luminous disk. Since the system uses only 120 degrees of the circular portion of the moon's limb to generate the tracking inputs, this analysis is valid for all phases of the moon irrespective of the rest of the luminous contour. Fabrication and Assembly of Electronics The electronics associated with the lunar tracker breadboard comprise a) preamplifier and signal shaper, b) data processor and computer, and c) servo amplifiers and readouts. In the fabrication of this design Northrop has elected to use purchased subassemblies and solid-state devices to the greatest extent possible. Furthermore, with

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Fig. 8. Typical observatory recording of 3.6-day old moon.

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Expanded C Scan - - - - - '

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Hoon Li.mb (Stop Clock)

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A.B.C. Scan Lines

Strobe Pulse (Start Clock) I Logic

(Scale .1 c~Olt)

Fig. 9. Typical oscilloscope photographs of 3.6-day old moon taken in the observatory.

364

the exception of the servo amplifiers and power supply, which are packaged by the respective vendors, all circuits are on cards of similar size and dimensions. This technique has minimized the effort spent in detailed design and component fabrication and checkout, and has correspondingly maximized reliability.

sightline generated by the tracker breadboard. In this experiment the instrument was assembled on an equatorial mount having a massive pedestal. The tracker sub-base was aligned with the eq~torial adjustments and checked with a sub-base-mounted finder telescope. The assembly was aligned by observing a bright star through finder and boresight telescopes. The reticle assembly was rotated to align the line of cusps. The tracking error is measured from the servo loop. The alignment of the tracking axis with the center of curvature of the bright limb was observed with the boresight telescope. During the waning period of the moon in the latter part of February, 1962, and the waxing period during early March tracking tests were made every clear morning or evening. The maximum brightness of the early morning or evening sky against which tracking was successfully accomplished was approximately 15 foot-Lamberts, although by adjusting the Schmitt firing level to be above the rising sky noise, tracking could have continued through considerably brighter skies.

The preamplifier and associated circuits to change voltage levels and match impedances are assembled on a card mounted in the front of the tracker head. The wiring has been brought around the gimbals b~flexible leads since the angular movement about each axis is so limited. The signal shaper and digital computer are assembled on cards and racked into the connector panels, about 400 connections per panel. Appropriate test points are brought out to two sets of 20 pin jacks mounted in strips. All of the cards are inserted from the front and can be checked out through the series of pin jacks located on strips which are accessible from the rear.

A typical chart record appears in Fig. 8. Four signal channels and a timing channel are shown with the earlier part of the signal trace at the bottom of the chart. The first signal channel records the X-axis error signal as it leaves the computer. The spread of the data indicates the resolution at that point in the system. The alignment of the data with the steady state line at the top demonstrates correct tracking without bias. The second signal channel records the angle readout. Notice that the servo system has completely smoothed the data. In this experiment the lunar image was centered optically and the declination circle clamped. The hour circle was driven by the sidereal clock. Since no guiding was used to compensate for the movement of the moon in right ascension, the breadboard automatically followed the moon and generated the slanting trace. The third signal records the Y-axis error signal as it leaves the computer. The spread of data indicates the corresponding resolution of Yaxis information. The data out of the computer demonstrate correct tracking except for a 25second bias, probably caused by imperfect alignment of the line of cusps. The fourth channel records the angle read out on the Y-axis. The data are well smoothed, the bias in channel 3 has disappeared and the final output demonstrates correct tracking.

The servo amplifiers are vendor-supplied shelf items but adjusted to match the servo motors. Each unit is a multi-stage transistorized amplifier with adjustable gain of one thousand or less. These amplifiers are mounted behind the third panel in the electronics cabinet which is shown in Fig. 7. Testing Program The program for testing the experimental model and appraising its performance consisted of two phases: laboratory observation of a simulated moon, and observatory measurements of the natural moon. Tests were performed in the laboratory using a simulated moon made up of a luminous disk imaged at infinity by means of a large collimator. Diaphragms were provided to simulate changes in range and in phase. Using a proper eyepiece reticle the boresight system was used to align the tracker visually and this alignment supplied a zero reference for studying tracker errors. During some of the tests the simulated lunar image was standardized at 900 foot-Lamberts and 6000 degrees K color temperature to closely approximate the photometric properties of the natural moon. With the 30-inch objective, simulated lunar images 0.5 and 1.9 degrees in diameter were successfully tracked. When the terminator was roughened to simulate moon craters, the computer failed to function properly. A circuit was designed and built into the computer which prevented these extra on-off signals from introducing erroneous information and a repetition of the tests with the simulated lunar craters showed that the difficulty was corrected, and satisfactory tracking was re-established. The laboratory tests indicated the tracking error to be less than 20 seconds with respect to both reproducibility and stability.

The wave forms of the signal in the computer are shown in Fig. 9. The oscilloscope pattern serves to monitor the system performance and to measure the signal-to-noise ratio. Traces of the A, B, and C channels are shown separately and the A-B-C sequence is shown as a single trace. The significant features are indicated by the callouts on the figure. Table 4 summarizes the data derived from experiments in which the natural moon was tracked from the Northrop observatory.

The philosophy of the observatory test program is to compare, by experiment, the true sightline to the natural moon with the 365

y

y

I

Instrument Axis

I

I---

;I

Moon Coordinates

/

Limb of Moon

Fig. 10. Geometry of many-look scanner.

Mirror

Spiral Slit

Radial Slit

/ Fig. 11. Optical diagram of many-look scanner.

366

TABLE 4

and is equivalent to an angular increment at the moon.

SUMMARY OF MOON TRACKING EXPERIMENTS

Since the measuring stations must be disposed in a definite manner with respect to the orientation of the lunar countour, correlation must be established with the coordinate system fixed in the instrument so that the scanning reticle pattern can be rotated into the correct relationship with the sun line. The instrument coordinate system is identified by the gimbal axes OX and OY. The two systems are related by the angle ~.

SiIMl-to·Nohe

.....

Hoon Phase 6. (days)

Date (1962)

21 F.b

22 wane

28 Feb

23 wane

Track or No Track

T or NI'

Ratio (Hinll8um)

11: 1

8,1

Notes Adjusted Hn. of cusp.

EarlY trackin-: hampered by atmospheric. lAter hampered by day11sht. extinction.

1 Mar

24 wane

8 Har

1.6 wane

9 Mar

2.6 wane

7.5:8

Hr

TO-3 min)

Approx. 2:1 2:1

Stability UO arc-sec, Pitch and yaw clock drive off. Temp 3S·P.

The scanning equations identify the error in the position of the lunar center with respect to the instrument center. Two somewhat different approaches have been considered. The first uses a least squares approach, has definite defense against random errors but requires digital multiplication by a large number of factors. The second approach, designed to minimize digital multiplication is based on the series expansion of an exact SOlution. It is not deliberately defensive against random errors, but may be adequately defensive in fact.

Switch to .....xina phue Aho tracked .un

10 Har

3.6 wax

T(up co 17 min)

10:1

11 Har

4.6 wax

T(up to 20 mln)

16: 1

8 arc-min track fie id

170: 1

Stronleat sl!laal •. Start pulse •• turatea. lntermlttant f08·

Max. a1av of mount ok

Track field halllaller

14.6 wane (near full)

21 Har

DESIGN OF A MANY-SPOKE PLANET TRACKER The validity of the radial measurements approach to planet tracking has been demonstrated by the experimental tracking of the moon with a three-spoke instrument. In the further development of this technique for tracking a partially illuminated body several improvements are desirable. These include observations at frequent intervals along the bright limb, progressive changes in the field-of-view to facilitate tracking in the vicinity of a body where the body subtends a large angle at the instrument. These design improvements are discussed in the following paragraphs.

Conceptual Design of the Instrument The optical and mechanical schematic of the many-look tracker is shown in Fig. 11. In the configuration shown two features are of interest. The many-look scanner concept just discussed from a logical point of view is presented. Another design feature, the tiltable head prism, permits the observation of planetary disks over a very wide range of angular subtense. Typical extreme values of half angle are 4-1/2 and 70 degrees. The corresponding ranges to the surface of the moon are 12,600 mi. and 67 mi. To the surface of the earth the ranges for these angles are 46,400 mi. and 257 mi. respectively.

Logical Design of the Scanning Subsystem The planetary tracker senses center of the earth or moon by computations based on measurements of the circular portion of the limb contour. These data are obtained from short radial segments of scan spaced in a conical annulus developed by rotating a suitable optical element in the instrument. This instrument design is applicable to any partially illuminated celestial body but as a convenience of expression the discussion will be either general or related to lunar observations.

As the figure shows, in order to generate these large sighting angles to the bright limb of a nearby body the device scans a conical field of half angle cl> by means of a special head prism tilted at the angle cI>/2. The sightline is defined by the objective and the optical axis XX'. The prism tilt angle cI>/2 is adjustable. In using the instrument the prism is rotated at constant angular velocity about the axis XX'. The detailed scan of the instantaneous field is accomplished by two reticles mounted close together in the focal plane and the diameter of the two reticle patterns defines the instantaneous field of view. The engineering details of this part of the instrument and related logic are discussed below.

The geometry of the scanning and sensing scheme is shown in Fig. 10. The insert shows the conical annulus which is 2 degrees wide and defines the outer half cone angle. Consider a lunar orthogonal coordinate system such that the sun lies on the positive extension of the Ox-axis. The circular limb of the moon will be convex to the right. The instrument axis is identified by the center 0, or by the concentric field stop circle which is shown. Radial offsets between the limb of the moon and the instrument circle are shown at 5-degree intervals and the magnitude of an offset is symbolized by y .. The offset on the x-axis (sun line) is identified as y. The consecutive measuring points extend symmgtrically from y at 5-degree intervals. The magnitude of any y.ois a linear measurement in the focal plane

When the sightline deviation angle is quite small a first surface head mirror would operate near grazing incidence and would be impractically large. Consequently a Dove prism which is dimensionally compact and is functionally the equivalent of a mirror is substituted. The prism is pivoted at the apex and is large enough to avoid vignetting the lens objective under any condition. This prism is shown in Fig. 11. As the range from the instrument to the planet changes the angular subtense will change and require progressive adjustment of Dove prism angle.

l

367

~------3.00--------i

Fig. 12. Design of scanning reticles (dimensions in inches).

Values of r i

COUNTER

SYN~~~~NIZED RETICLE

1-----., "-,-----,,.--'

MOON TO GIMBAL COORDINATES CONVERTER

EX SERVe

E. Y SERVO

COMMAND

COMMAND

Fig. 13. Block diagram of computing logic.

368

1J - R SERVO COMMAND

Typically this slewing rate is about 0.4 deg/sec of time. The present design uses a telephoto objective with 25 inch focal length. Consequently linear distances in the focal plane correspond to angular distances in object space in the ratio of 135 arcmin/inch. The telephoto design will be as conservative as possible without sacrificing the necessary instrument compactness. The optical specifications for the objective are as follows: Equivalent focal length Front vertex to focal plane Clear aperture of front lens Relative aperture Field of View Type of achromatization

25 inches 16-112 inch maximum 1.35 inches F/5.4 2-112 degrees, half field D, F and G'

The scanning mechanism is based on the same principles discussed earlier, but the equipment is modified to permit many scans along the bright lim~ In the scan routine a small spot sweeps an annulus in a series of equi-spaced radial lines by means of the intersection of a spiral slit and a radial slit mounted in the focal plane. Their dimensions are shown in Fig. 12. The radial slit turns in exact synchronism with the Dove prism and is so phased that the slit remains in the principal plane of the prism. As the Dove prism rotates, the slit sweeps an annulus which is about 2 degrees wide and has a radius dependent on the selection of the Dove prism angle q,. In order to sample the annulus in a series of discrete radial scans, and to avoid excessive reticle speeds, three spirals are used. In a single scanning sequence the radial portion of one of the spiral patterns in the fast reticle lines up for an instant with the radial slit of the slow reticle. As the radial slit elements move out of coincidence the spiral and radial slits intersect in a rectangular aperture which moves at constant linear velocity from the rim toward the center of the field. With three spirals on the reticle, a gear ratio of 23:1 and counter-rotation between the two reticles, a radi~l line is scanned every 5 degrees and there are 72 discrete looks in a complete horizon circle. The instantaneous field is 1.4 x 0.7 arc min. in size. Selecting a photomultiplier with S-ll surface as the most desirable sensor, and selecting reasonable values for the other radiometric parameters, the spectral response of the system will extend from 0.38 to 0.68 microns with the peak at about 0.5 microns. This spectral distribution indicates the desirability of achromatizing the objective lens for the wavelengths corresponding to the Fraunhofer lines D, F and G'.

the flux onto the photocathode not withstanding the heavy spherical and chromatic aberrations inherent in so simple an approach. The data processing function includes those operations relating to the signal in the photocell, the logic and computations, and finally the servo loops which minimize errors in sightline and range angles. Since the bright limb of the moon is a semicircle, symmetrical with respect to the sun-moon line, the related limb measurements naturally refer to a lunar coordinate system using the sun line as one axis of reference. In the course of a tracking exercise this lunar coordinate set is rotated through an angle ~ with respect to the instrument coordinates identified with the X and Y gimbals. The angle ~ ranges through 360 degrees depending on the relative position of sun, moon and sighting instrument; the latter being on earth or in a space vehicle as the case may be. The scanning operation involves 32 looks which are 5 degrees apart and which embrace 160 degrees of arc. The initiation of each look is signalled by a synchronizing pulse generated by a magnetic mark on the reticle rim and the readout head which correlates the moon and the instrument coordinate sets. The photocell signal resul ts from that JXJrtion of the lunar image flux which passes through the intersecting portions of the radial slit and the spiral slit reticles. If the lunar image is within the field of view, a portion will be intercepted by the aperture of the intersecting slits in a manner characteristic of the situation. A considerable portion of the lunar flux will be transmitted by each coincidence of the radial slit elements as the fast reticle pattern rotates past the slow one. Typically the intersection of radial and spiral slits will cross onto the bright lunar image after an interval representative of the spacing between the lunar contour and the rim of the tracker field of view. The spacing between these pulses measures the radial interval between the bright limb and the edge of the instrumental field stop. By auxiliary logic various special cases can be considered such that the partially illuminated body can be tracked under all conditions. The computer logic is shown as a block diagram in Fig. 13. Notice that the outputs include not only two sightline angles, but range also.

CONCLUSION This paper has discussed two closely related techniques for automatically tracking a partially illuminated body such as a planet or the mOon. The sightline to the center is determined from many measurements spaced along the circular bright limb. Means are described by which the change in angular subtense can be controlled. This technique contributes significantly to the methodology by which spacecraft can be navigated between earth, the moon, and the planets.

The field lens is a relatively thick planoconvex singlet fabricated to condenser quality. Functionally it accepts the ray bundle defined by the objective and the full length of the radial slit and redirects the flux into the aperture of the sensor. The focal length and other design parameters were selected to effectively project

369

ACKNOWLEDGEMENT

D. Van Nostrand, Inc., 1960, p 191.

The information described in this paper resulted from studies performed at Northrop Space Laboratories. The work was supported to a major degree by the Aeronautical Systems Division, Air Force Systems Command, U. S. Air Force. The support of these organizations is appreciated.

REFERENCES (1)

(2)

Pettit, E. and Nicholson, S. B., "Lunar Rad iations and Temperatures ," Astrophysical Journal ll, 102-135, 1930. Ehricke, K. A., "Space Flight, 1. Environment and Celestial Mechanics," New York:

370

(3)

Fielder, Gilbert, "Structure of the Moon r s Surface," New York: Pergamon Press, 1961, pg 47.

(4)

Allen, C. W., "Astrophysical Quantities," London: Athlone Press, 1955, pg 24.

(5)

Ref. 4 pp 24, 162.

(6)

United States Air Force, "Handbook of Geophysics," New York: Macmillan, 1960, Chap. 2.

(7)

Ref. 4, various pages.

(8)

Kuiper, Gerard P., Ed., "The Earth as a Planet," Chicago: U. of Chicago Press, 1954, pp 733-734.